Identifiability of homogeneous polynomials and Cremona Transformations

TL;DR

This paper characterizes when a general homogeneous polynomial has a unique decomposition into powers of linear forms by classifying certain Cremona transformations, resolving a long-standing problem in algebraic geometry.

Contribution

It provides a complete classification of degrees and dimensions where general polynomials are identifiable, linking the problem to Cremona transformations.

Findings

Identifies all (d, n) pairs with identifiable polynomials.

Classifies a special class of Cremona transformations.

Completes a century-old classification problem.

Abstract

A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.